In this case, the subject could be either the speaker or the listener, but it's important to note that based on context, the conveyed meaning does not change for either subjects.
俺が始めてみるか (monologue, saying to self: should I start and see?)
お前、始めてみるか (saying to other: you! start and see?)
Either way, you know these two are gonna fight. The Japanese language is largely dependent on context and is inherently ambiguous. (Ex. 社長に紹介されます. Who is doing the introduction?) So, we have to always speak from context when looking at meanings of sentences. In this case, the conveyed meaning does not change for either subject. If the meanings were to change with the subject, then people can deduce which one is the intended meaning from the context. Since there are two different meanings(unlike this case), one must be much more likely over the other from context. If both are equally likely, then the speaker must re-word something to clear the ambiguity. Just like in the example 社長に紹介されます.
社長に紹介されました I was introduced to the boss (by someone).
社長に紹介されました I was introduced by the boss (to someone).
Here, without more sentences to give more context, two meanings are equally likely. But with
始めてみるか and the context that
opponents with swords about to start a battle, there is no ambiguity at all. Who's the exact subject then, you ask? Sorry, I don't think that even a native speaker can answer that question for you, as the Japanese language is inherently ambiguous.